Abstrakt
A graph G is called a sum graph if there is a so-called sum labeling of G, i.e. an injective function l : V(G) -> N such that for every u, v is an element of V (G) it holds that uv is an element of E(G) if and only if there exists a vertex w is an element of V (G) such that l(u) + l(v) = l(w). We say that sum labeling l is minimal if there is a vertex u is an element of V (G) such that l(u) = 1.
In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth.